A FEniCS Tutorial - FEniCS Project

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numerical u at the center point: [ 1.83333333] exact u at the center point: [ 1.75] The discrepancy is due to the fact that the center point is not a node in this particular mesh, but a point in the interior of a cell, and u varies linearly over the cell while u0 is a quadratic function. We have seen how to extract the nodal values in a numpy array. If desired, we can adjust the nodal values too. Say we want to normalize the solution such that max j U j = 1. Then we must divide all U j values by max j U j . The following snippet performs the task: max_u = u_array.max() u_array /= max_u u.vector()[:] = u_array u.vector().set_local(u_array) # alternative print u.vector().array() That is, we manipulate u_array as desired, and then we insert this array into ‘u‘’s Vector object. The /= operator implies an in-place modification of the object on the left-hand side: all elements of the u_array are divided by the value max_u. Alternatively, one could write u_array = u_array/max_u, which implies creating a new array on the right-hand side and assigning this array to the name u_array. A call like u.vector().array() returns a copy of the data in u.vector(). Onemustthereforeneverperformassignmentslikeu.vector.array()[:] = ..., but instead extract the numpy array (i.e., a copy), manipulate it, and insert it back with u.vector()[:] = or u.set_local(...). All the code in this subsection can be found in the file d4_p2D.py in the stationary/poisson directory. We have commented out the plotting statements in this version of the program, but if you want plotting to happen, make sure that interactive is called at the very end of the program. 1.7 Solving a Real Physical Problem Perhaps you are not particularly amazed by viewing the simple surface of u in the test problem from Section 1.3. However, solving a real physical problem with a more interesting and amazing solution on the screen is only a matter of specifying a more exciting domain, boundary condition, and/or right-hand side f. One possible physical problem regards the deflection D(x,y) of an elastic circular membrane with radius R, subject to a localized perpendicular pressure force, modeled as a Gaussian function. The appropriate PDE model is −T∇ 2 D = p(x,y) in Ω = {(x,y)|x 2 +y 2 ≤ R}, (14) with p(x,y) = A 2πσ exp (− 1 2 ( ) 2 x−x0 − 1 ( ) ) 2 y −y0 . (15) σ 2 σ 20
Here, T is the tension in the membrane (constant), p is the external pressure load, A the amplitude of the pressure, (x 0 ,y 0 ) the localization of the Gaussian pressure function, and σ the ”width” of this function. The boundary of the membrane has no deflection, implying D = 0 as boundary condition. For scaling and verification it is convenient to simplify the problem to find an analytical solution. In the limit σ → ∞, p → A/(2πσ), which allows us to integrate an axi–symmetric version of the equation in the radial coordinate r ∈ [0,R] and obtain D(r) = (r 2 − R 2 )A/(8πσT). This result gives a rough estimate of the characteristic size of the deflection: |D(0)| = AR 2 /(8πσT), which can be used to scale the deflecton. With R as characteristic length scale, we can derive the equivalent dimensionless problem on the unit circle, −∇ 2 w = f, (16) with w = 0 on the boundary and with ( f(x,y) = 4exp − 1 ( ) 2 Rx−x0 − 1 ( ) ) 2 Ry −y0 . (17) 2 σ 2 σ For notational convenience we have dropped introducing new symbols for the scaled coordinates in (17). Now D is related to w through D = AR 2 w/(8πσT). Let us list the modifications of the d1_p2D.py program that are needed to solve this membrane problem: • Initialize T, A, R, x 0 , y 0 , and σ, • create a mesh over the unit circle, • make an expression object for the scaled pressure function f, • define the a and L formulas in the variational problem for w and compute the solution, • plot the mesh, w, and f, • write out the maximum real deflection D. Some suitable values of T, A, R, x 0 , y 0 , and σ are T = 10.0 # tension A = 1.0 # pressure amplitude R = 0.3 # radius of domain theta = 0.2 x0 = 0.6*R*cos(theta) y0 = 0.6*R*sin(theta) sigma = 0.025 A mesh over the unit circle can be created by 21
Page 1 and 2: A FEniCS Tutorial Hans Petter Langt
Page 3 and 4: 7 Miscellaneous Topics 82 7.1 Gloss
Page 5 and 6: that changing the PDE and boundary
Page 7 and 8: The proper statement of our variati
Page 9 and 10: """ FEniCS tutorial demo program: P
Page 11 and 12: polynomial approximations over each
Page 13 and 14: Instead of nabla_grad we could also
Page 15 and 16: solution during program development
Page 17 and 18: The demo program d2_p2D.py in the s
Page 19: All mesh objects are of type Mesh s
Page 23 and 24: """ FEniCS program for the deflecti
Page 25 and 26: Figure 3: Plot of the deflection of
Page 27 and 28: Figure 4: Example of visualizing th
Page 29 and 30: Let us continue to use our favorite
Page 31 and 32: 1.11 Computing Functionals After th
Page 33 and 34: e_Ve.vector()[:] = u_e_Ve.vector().
Page 35 and 36: It is possible to restrict the inte
Page 37 and 38: Other functions for visualizing 2D
Page 39 and 40: 1 0.8 Contour plot of u 4 3.5 3 2.5
Page 41 and 42: accordingly: f = −6, { −4, y =
Page 43 and 44: Here, Γ 0 is the boundary x = 0, w
Page 45 and 46: The object A is of type Matrix, whi
Page 47 and 48: with exact solution u(x) = x 2 . Ou
Page 49 and 50: The variational formulation of our
Page 51 and 52: iter += 1 solve(a == L, u, bcs) dif
Page 53 and 54: tol = 1E-14 def left_boundary(x, on
Page 55 and 56: We may collect the terms with the u
Page 57 and 58: as ǫ → 0. This last expression i
Page 59 and 60: 3.1 A Diffusion Problem and Its Dis
Page 61 and 62: • if u 1 is to be computed by pro
Page 63 and 64: while t 1, but for the values of N
Page 65 and 66: if we identify the matrix M with en
Page 67 and 68: y T 0 (t) = T R + T A sin(ωt) x D
Page 69 and 70: straightforward approach is to defi
Page 71 and 72:
def T_exact(x): a = sqrt(omega*rho*
Page 73 and 74:
Theta = pi/2 a, b = 1, 5.0 nr = 10
Page 75 and 76:
y ✻ u = 1 Ω 1 ∂u ∂n = 0 ∂u
Page 77 and 78:
subdomain1 = Omega1() subdomain1.ma
Page 79 and 80:
The weak form then becomes ∫ ∫
Page 81 and 82:
A = assemble(a, exterior_facet_doma
Page 83 and 84:
7.2 Overview of Objects and Functio
Page 85 and 86:
Parameters in expression strings mu
Page 87 and 88:
} "smoother: type" : "ML symmetric
Page 89 and 90:
2. forgetting that the spatial coor
Page 91 and 92:
Applied Mathematics (SIAM), Philade
Page 93 and 94:
Index alg newton np.py, 52 assemble
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A FEniCS Tutorial - FEniCS Project

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